Robin bought a computer for $1,250. It will depreciate, or decrease in value, by 10% each year that she owns it.a) Is the sequence formed by the value at the beginning of each year arithmetic, geometric, or neither? b) Write an explicit formula to represent the sequence. c) Find the value of the computer at the beginning of the 6-th year.

brizgney58z

brizgney58z

Answered question

2023-02-21

Robin bought a computer for $1,250. It will depreciate, or decrease in value, by 10% each year that she owns it.a) Is the sequence formed by the value at the beginning of each year arithmetic, geometric, or neither? Explain.b) Write an explicit formula to represent the sequence.c) Find the value of the computer at the beginning of the 6thyear.

Answer & Explanation

Marquise Cole

Marquise Cole

Beginner2023-02-22Added 3 answers

Discover the order created by the arithmetic, geometric, or neither value at the start of each year.
Let's locate the sequence's first three terms.
Since the first term of the sequence is a1=1250.
Since the value will depreciate, or decrease in value, by 10% each year that she owns it
The second term of the sequence:
Subtract 10%of the value from the original value 1250.
1250of10%=1250×10100=125
Subtract 125 from the first term:
1250-125=1125
Find the third term of the sequence:
Subtract 10%of the value from the second value 1125.
1125of10%=1125×10100=112.5
Subtract 125 from the second term:
1125-112.5=1012.5
Therefore, the first three terms of the sequence will be 1250,1125,and1012.5.
The ratio r to prove the sequence is arithmetic, geometric, or neither:
Substitute an-1=1250andan=1125:
r=anan-1r=11251250r=0.9
Substitute an-1=1125andan=1012.5:
r=anan-1r=1012.51125r=0.9
Recall the definition of a geometric sequence, which is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by, a fixed non-zero number called the common ratio.
Therefore, the sequence is formed by the geometric sequence.
Write an explicit formula to represent the sequence.
The general formula for the geometric sequence is,
an=a1rn-1
Substitute a1=1250andr=0.9:
an=a1rn-1an=12500.9n-1
Therefore, the required explicit formula is an=(1250)(0.9)n-1.
Find the value of the computer at the beginning of the 6th year.
The explicit formula for the geometric sequence at n=6 is,
an=(1250)(0.9)n-1a6=(1250)(0.9)6-1a6=(1250)(0.9)5a6=$738.1125
Therefore, the value of the computer at the beginning of the 6th year will be $738.1125.

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